Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Large time behavior of solutions to the generalized damped wave equation ${\displaystyle u_{tt}+A{u}_t+\nu Bu+F(x,t,u,u_t,\nabla u)=0}$ for ${\displaystyle (x,t)\in {ℝ}^n\times [0,\infty )}$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where ${\displaystyle A{u}_t=u_t}$, ${\displaystyle Bu=-\Delta u}$, and the nonlinear term is either ${\displaystyle {\left|u_t\right|}^{q-1}{u}_t}$ or ${\displaystyle {\left|u\right|}^{\alpha -1}u}$. In this case, the asymptotic profile of solutions is given by a multiple of the Gauss-Weierstrass kernel. Our method of proof does not require the smallness assumption on the initial conditions.